Whakaoti i te {l}{-(-x-y)-4(y-x)=8}{3x-1+2(y+3)-5=20}

$$\left\{ \begin{array} { l } { - ( - x - y ) - 4 ( y - x ) = 8 } \\ { 3 x - 1 + 2 ( y + 3 ) - 5 = 20 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Hei kimi i te tauaro o -x-y, kimihia te tauaro o ia taurangi.

$$-\left(-x\right)+y-4\left(y-x\right)=8$$

Whakamahia te āhuatanga tohatoha hei whakarea te -4 ki te y-x.

$$-\left(-x\right)+y-4y+4x=8$$

Pahekotia te y me -4y, ka -3y.

$$-\left(-x\right)-3y+4x=8$$

Whakareatia te -1 ki te -1, ka 1.

$$x-3y+4x=8$$

Pahekotia te x me 4x, ka 5x.

$$5x-3y=8$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te y+3.

$$3x-1+2y+6-5=20$$

Tāpirihia te -1 ki te 6, ka 5.

$$3x+5+2y-5=20$$

Tangohia te 5 i te 5, ka 0.

$$3x+2y=20$$

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$5x-3y=8,3x+2y=20$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te x mā te wehe i te x i te taha mauī o te tohu ōrite.

$$5x-3y=8$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$5x=3y+8$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(3y+8\right)$$

Whakareatia \frac{1}{5} ki te 3y+8.

$$x=\frac{3}{5}y+\frac{8}{5}$$

Whakakapia te \frac{3y+8}{5} mō te x ki tērā atu whārite, 3x+2y=20.

$$3\left(\frac{3}{5}y+\frac{8}{5}\right)+2y=20$$

Whakareatia 3 ki te \frac{3y+8}{5}.

$$\frac{9}{5}y+\frac{24}{5}+2y=20$$

Tāpiri \frac{9y}{5} ki te 2y.

$$\frac{19}{5}y+\frac{24}{5}=20$$

Me tango \frac{24}{5} mai i ngā taha e rua o te whārite.

$$\frac{19}{5}y=\frac{76}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{19}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=4$$

Whakaurua te 4 mō y ki x=\frac{3}{5}y+\frac{8}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{5}\times 4+\frac{8}{5}$$

Whakareatia \frac{3}{5} ki te 4.

$$x=\frac{12+8}{5}$$

Tāpiri \frac{8}{5} ki te \frac{12}{5} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=4$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Hei kimi i te tauaro o -x-y, kimihia te tauaro o ia taurangi.

$$-\left(-x\right)+y-4\left(y-x\right)=8$$

Whakamahia te āhuatanga tohatoha hei whakarea te -4 ki te y-x.

$$-\left(-x\right)+y-4y+4x=8$$

Pahekotia te y me -4y, ka -3y.

$$-\left(-x\right)-3y+4x=8$$

Whakareatia te -1 ki te -1, ka 1.

$$x-3y+4x=8$$

Pahekotia te x me 4x, ka 5x.

$$5x-3y=8$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te y+3.

$$3x-1+2y+6-5=20$$

Tāpirihia te -1 ki te 6, ka 5.

$$3x+5+2y-5=20$$

Tangohia te 5 i te 5, ka 0.

$$3x+2y=20$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$5x-3y=8,3x+2y=20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-3\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\20\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-3\\3&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-3\\3&2\end{matrix}\right))\left(\begin{matrix}5&-3\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-3\\3&2\end{matrix}\right))\left(\begin{matrix}8\\20\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-3\\3&2\end{matrix}\right))\left(\begin{matrix}8\\20\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-3\\3&2\end{matrix}\right))\left(\begin{matrix}8\\20\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right) te poukapa kōaro, nō reira ka taea te tuhi anō te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5\times 2-\left(-3\times 3\right)}&-\frac{-3}{5\times 2-\left(-3\times 3\right)}\\-\frac{3}{5\times 2-\left(-3\times 3\right)}&\frac{5}{5\times 2-\left(-3\times 3\right)}\end{matrix}\right)\left(\begin{matrix}8\\20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{19}&\frac{3}{19}\\-\frac{3}{19}&\frac{5}{19}\end{matrix}\right)\left(\begin{matrix}8\\20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{19}\times 8+\frac{3}{19}\times 20\\-\frac{3}{19}\times 8+\frac{5}{19}\times 20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=4$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Hei kimi i te tauaro o -x-y, kimihia te tauaro o ia taurangi.

$$-\left(-x\right)+y-4\left(y-x\right)=8$$

Whakamahia te āhuatanga tohatoha hei whakarea te -4 ki te y-x.

$$-\left(-x\right)+y-4y+4x=8$$

Pahekotia te y me -4y, ka -3y.

$$-\left(-x\right)-3y+4x=8$$

Whakareatia te -1 ki te -1, ka 1.

$$x-3y+4x=8$$

Pahekotia te x me 4x, ka 5x.

$$5x-3y=8$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te y+3.

$$3x-1+2y+6-5=20$$

Tāpirihia te -1 ki te 6, ka 5.

$$3x+5+2y-5=20$$

Tangohia te 5 i te 5, ka 0.

$$3x+2y=20$$

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$5x-3y=8,3x+2y=20$$

Kia ōrite ai a 5x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$3\times 5x+3\left(-3\right)y=3\times 8,5\times 3x+5\times 2y=5\times 20$$

Whakarūnātia.

$$15x-9y=24,15x+10y=100$$

Me tango 15x+10y=100 mai i 15x-9y=24 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x-9y-10y=24-100$$

Tāpiri 15x ki te -15x. Ka whakakore atu ngā kupu 15x me -15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-9y-10y=24-100$$

Tāpiri -9y ki te -10y.

$$-19y=24-100$$

Tāpiri 24 ki te -100.

$$-19y=-76$$

Whakawehea ngā taha e rua ki te -19.

$$y=4$$

Whakaurua te 4 mō y ki 3x+2y=20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+2\times 4=20$$

Whakareatia 2 ki te 4.

$$3x+8=20$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$3x=12$$

Whakawehea ngā taha e rua ki te 3.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=4$$